Local well-posedness and blow-up of solution for a higher-order wave equation with viscoelastic term and variable-exponent

We investigate in this paper a value problem related to the following nonlinear higher-order wave equation     \eta_{tt}+\left(  -\Delta\right)  ^{m}\eta-%    %TCIMACRO{\dint \limits_{۰}^{t}}%    %BeginExpansion    {\displaystyle\int\limits_{۰}^{t}}    %EndExpansion    g\left(  t-s\right)  \left(  -\Delta\right)  ^{m}\eta\left(  s\right)    ds+\eta_{t}=\left\vert \eta\right\vert ^{p\left(  x\right)  -۲}\eta.   Firstly, we prove the existence and uniqueness of the local solution under suitable conditions for the relaxation function g and viable-exponent p\left(  .\right)  , using a method, which is a mixture of the Faedo-Galarkin and Banach fixed point theorem, and prove also the solution blows up in finite time. Finally, we give a two-dimensional numerical example to illustrate the blow-up result.

کلیدواژه ها:

Higher-order equation ، Wave Equation ، variable-exponent ، local solution ، Blow up